Newton's Method

In the applet below, enter (leaving no blanks) the coefficients of the
polynomial p(x) and the endpoints of an interval [a,b] in which you want to
find a root. Pressing the "Plot" button will draw the graph of p(x). Then
enter a first approximation x_1 of a root for p(x) in [a,b]. (For a nice picture, choosing x_1 rather far from the root is generally best.) For any
point x_n, Newton's method (also called the Newton-Raphson method) calculates

x_(n+1) = x_n - p(x_n)/p'(x_n).

Geometrically, x_(n+1) is where the tangent line to the graph at
(x_n,p(x_n)) crosses the x-axis, and is usually a better approximation of the
root than x_n was. Press the "Newton step" button repeatedly until you get
an approximation with the desired accuracy. If you now press the "Plot"
button, the graph will be cleared and redrawn, reflecting any changes you made
in the coefficients or endpoints.